| L(s) = 1 | − 2·2-s − 6.24·3-s + 4·4-s + 12.2·5-s + 12.4·6-s + 7·7-s − 8·8-s + 11.9·9-s − 24.4·10-s + 46.9·11-s − 24.9·12-s + 38.8·13-s − 14·14-s − 76.4·15-s + 16·16-s − 77.4·17-s − 23.9·18-s − 42.2·19-s + 48.9·20-s − 43.6·21-s − 93.9·22-s + 23·23-s + 49.9·24-s + 24.8·25-s − 77.6·26-s + 93.8·27-s + 28·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.20·3-s + 0.5·4-s + 1.09·5-s + 0.849·6-s + 0.377·7-s − 0.353·8-s + 0.443·9-s − 0.774·10-s + 1.28·11-s − 0.600·12-s + 0.828·13-s − 0.267·14-s − 1.31·15-s + 0.250·16-s − 1.10·17-s − 0.313·18-s − 0.510·19-s + 0.547·20-s − 0.454·21-s − 0.910·22-s + 0.208·23-s + 0.424·24-s + 0.199·25-s − 0.585·26-s + 0.668·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.144004504\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.144004504\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 7 | \( 1 - 7T \) |
| 23 | \( 1 - 23T \) |
| good | 3 | \( 1 + 6.24T + 27T^{2} \) |
| 5 | \( 1 - 12.2T + 125T^{2} \) |
| 11 | \( 1 - 46.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 38.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 77.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 42.2T + 6.85e3T^{2} \) |
| 29 | \( 1 + 105.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 57.1T + 2.97e4T^{2} \) |
| 37 | \( 1 - 93.9T + 5.06e4T^{2} \) |
| 41 | \( 1 - 8.41T + 6.89e4T^{2} \) |
| 43 | \( 1 - 413.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 539.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 414.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 55.9T + 2.05e5T^{2} \) |
| 61 | \( 1 - 538.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 73.6T + 3.00e5T^{2} \) |
| 71 | \( 1 + 213.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 653.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 437.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 507.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 871.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 284.T + 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.10147681810895569245666095799, −10.44185108044189751925656447739, −9.274983629287338075516566963250, −8.695502352813265335562010639319, −7.08515590800556863436641780977, −6.21184183411002498918917293745, −5.65721618198841209698728367657, −4.18393541210260094788796939093, −2.10193944353949068741460186696, −0.897732956005765290386522660885,
0.897732956005765290386522660885, 2.10193944353949068741460186696, 4.18393541210260094788796939093, 5.65721618198841209698728367657, 6.21184183411002498918917293745, 7.08515590800556863436641780977, 8.695502352813265335562010639319, 9.274983629287338075516566963250, 10.44185108044189751925656447739, 11.10147681810895569245666095799
